Trippy Triangles

We need our three numbers $a,b,c$ to satisfy the triangular inequalities:

$a+b>c$

$b+c>a$

$c+a>b$

We can see that none of the numbers can be equal to $1$ , since this would infer (setting $a=1$ ) that $b+1>c$ and $c+1>b$ which is impossible for distinct integers $b,c$ . The remaining combinations to check are:

$(2,3,4)$ : valid;

$(2,3,5)$ : invalid (this gives you a degenerate triangle, since $2+3=5$ );

$(2,4,5)$ : valid;

$(3,4,5)$ : valid.

Hence we are left with $\boxed{3}$ possible triangles.